Is the word cansado the same in Brazilian and European Portuguese?

Yes

The word "cansado" is identical in meaning, grammar, spelling, and typical usage in both Brazilian and Continental Portuguese. The only difference lies in pronunciation:

• Brazilian Portuguese: The vowels tend to be more "open." The "a" in the second syllable is pronounced clearly and openly (e.g., /kanˈsa.du/).
• Continental Portuguese: The vowels are more "reduced" or closed. The unstressed "a" often becomes a nearly silent or very short neutral sound (e.g., /kɐ̃ˈsɐ.du/).

Brazilian Portuguese Examples

1. Eu tô muito cansado hoje. (I am very tired today.)
2. A gente tá cansado dessa situação. (We are tired of this situation.)
3. Você tá cansado de tanto trabalhar? (Are you tired from working so much?)
4. Eu fiquei cansado depois da corrida. (I became tired after the run.)
5. Ele parece cansado, mas não diz nada. (He looks tired, but he doesn't say anything.)

Continental Portuguese Examples

1. Estou bastante cansado hoje. (I am quite tired today.)
2. Nós estamos cansados desta situação. (We are tired of this situation.)
3. Estás cansado de tanto trabalhar? (Are you tired from working so much?)
4. Fiquei cansado após a corrida. (I became tired after the run.)
5. Ele parece cansado, mas não diz nada. (He looks tired, but he doesn't say anything.)